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Theorem opeluu 4860
Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
opeluu.1 𝐴 ∈ V
opeluu.2 𝐵 ∈ V
Assertion
Ref Expression
opeluu (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))

Proof of Theorem opeluu
StepHypRef Expression
1 opeluu.1 . . . 4 𝐴 ∈ V
21prid1 4240 . . 3 𝐴 ∈ {𝐴, 𝐵}
3 opeluu.2 . . . . 5 𝐵 ∈ V
41, 3opi2 4859 . . . 4 {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵
5 elunii 4371 . . . 4 (({𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝐶)
64, 5mpan 701 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → {𝐴, 𝐵} ∈ 𝐶)
7 elunii 4371 . . 3 ((𝐴 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐴 𝐶)
82, 6, 7sylancr 693 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴 𝐶)
93prid2 4241 . . 3 𝐵 ∈ {𝐴, 𝐵}
10 elunii 4371 . . 3 ((𝐵 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝐵} ∈ 𝐶) → 𝐵 𝐶)
119, 6, 10sylancr 693 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 𝐶)
128, 11jca 552 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  Vcvv 3172  {cpr 4126  cop 4130   cuni 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367
This theorem is referenced by:  asymref  5418  asymref2  5419  wrdexb  13120
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